Symmetries of Convex Sets in the Hyperbolic Plane

Jesús Jerónimo-Castro; Francisco G. Jimenez-Lopez

doi:10.68381/jca26059

Abstract

We establish some results characterizing central or axial symmetry of convex sets in the hyperbolic plane. The characterizations follow the spirit of a Chakerian-Klamkin's characterization of central symmetry for Euclidean sets: if for any three point-subset M of a compact set K there is a symmetric image of M that is also contained in K, then K has a center of hyperbolic symmetry. We also study axial symmetry when the axis is either a geodesic, a horocycle, or a hypercycle. Finally, in the last section we give a characterization of the hyperbolic disc.

Author information

Contact details are reproduced from the original publication and may be historical.

Jesús Jerónimo-Castro

Facultad de Ingenieria, Universidad Autónoma de Querétaro, Querétaro, México

jesusjero@hotmail.com

Francisco G. Jimenez-Lopez

Facultad de Ingenieria, Universidad Autónoma de Querétaro, Querétaro, México

fjimenez@uaq.mx

Keywords

Hyperbolic disc
orthogonal symmetry
hyperbolic symmetry

Mathematics Subject Classification

52A10, 52A55

References

G. D. Chakerian, M. S. Klamkin: A three point characterization of central symmetry, Amer. Math. Montly 111 (2004) 903–905.https://doi.org/10.1080/00029890.2004.11920155 Google Scholar
L. Danzer, B. Grünbaum, V. Klee: Helly’s theorem and its relatives, in: Convexity, V. L. Klee (ed.), Proc. Symp. Pure Math. 7, American Mathematical Society, Providence (1963) 101–180.https://doi.org/10.1090/pspum/007/0157289 Google Scholar
B. V. Dekster: Double normals characterize bodies of constant width in Riemannian manifolds, in: Geometric Analysis and Nonlinear Partial Differential Equations, I. J. Bakelman (ed.), Lecture Notes Pure Appl. Math. 144, Marcel Dekker, New York (1993) 187–201.Google Scholar
J. Jerónimo-Castro, L. Montejano: Chakerian-Klamkin type theorems, J. Convex Analysis 17 (2010) 643–649.https://doi.org/10.68381/jca17042 Google Scholar
J. Jerónimo-Castro, S. Tabachnikov: Configuration spaces of plane polygons and a sub-Riemannian aproach to the equitangent problem, J. Dyn. Control Syst. 22 (2016) 227–250.https://doi.org/10.1007/s10883-015-9269-4 Google Scholar
L. Santaló: Note on convex curves on the hyperbolic plane, Bull. Amer. Math. Soc. 51 (1957) 405–412.https://doi.org/10.1090/S0002-9904-1945-08366-9 Google Scholar
L. Santaló: Convexity in the hyperbolic plane, Univ. Nac. Tucumán Rev. Ser. A 19 (1969) 173–183.Google Scholar

Suggested citation

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.

[1] G. D. Chakerian, M. S. Klamkin: A three point characterization of central symmetry, Amer. Math. Montly 111 (2004) 903–905.https://doi.org/10.1080/00029890.2004.11920155 Google Scholar

[2] L. Danzer, B. Grünbaum, V. Klee: Helly’s theorem and its relatives, in: Convexity, V. L. Klee (ed.), Proc. Symp. Pure Math. 7, American Mathematical Society, Providence (1963) 101–180.https://doi.org/10.1090/pspum/007/0157289 Google Scholar

[3] B. V. Dekster: Double normals characterize bodies of constant width in Riemannian manifolds, in: Geometric Analysis and Nonlinear Partial Differential Equations, I. J. Bakelman (ed.), Lecture Notes Pure Appl. Math. 144, Marcel Dekker, New York (1993) 187–201.Google Scholar

[4] J. Jerónimo-Castro, L. Montejano: Chakerian-Klamkin type theorems, J. Convex Analysis 17 (2010) 643–649.https://doi.org/10.68381/jca17042 Google Scholar

[5] J. Jerónimo-Castro, S. Tabachnikov: Configuration spaces of plane polygons and a sub-Riemannian aproach to the equitangent problem, J. Dyn. Control Syst. 22 (2016) 227–250.https://doi.org/10.1007/s10883-015-9269-4 Google Scholar

[6] L. Santaló: Note on convex curves on the hyperbolic plane, Bull. Amer. Math. Soc. 51 (1957) 405–412.https://doi.org/10.1090/S0002-9904-1945-08366-9 Google Scholar

[7] L. Santaló: Convexity in the hyperbolic plane, Univ. Nac. Tucumán Rev. Ser. A 19 (1969) 173–183.Google Scholar